We prove and give numerical results for two lower bounds and eleven upper bounds to the optimal constant k₀ = k₀(n,α) in the inequality ∥u∥_2n/(n-2α) ≤ k₀ ∥∇_u∥^α₂ ∥u∥^1-α ₂, u ∈ H¹(ℝⁿ), for n = 1, 0 < α ≤ 1/2, and n ≥ 2, 0 < α < 1. This constant k₀ is the reciprocal of the infimum λ_n,α for u ∈ H¹(ℝⁿ) of the functional Λ_n,α = ∥∇_u∥^α₂ ∥u∥^1-α ₂/∥u∥_2n/(n-2α), u ∈ H¹(ℝⁿ), where for n = 1, 0 < α ≤ 1/2, and for n ≥ 2, 0 < α < 1. The lowest point in the point spectrum of the Schrödinger operator τ = -Δ+q on ℝⁿ with the real-valued potential q can be expressed in λ_n,α for all q _ = max(0,-q) ∈ L^p(ℝⁿ), for n = 1, 1 ≤ p < ∞, and n ≥ 2, n/2 < p < ∞, and the norm ∥q _ ∥p.